Tayler instability of toroidal magnetic fields in MHD Taylor-Couette flows

TL;DR

This study investigates the Tayler instability in toroidal magnetic fields within Taylor-Couette flows, revealing how rotation, shear, and magnetic Prandtl number influence stability and angular momentum transport, with implications for laboratory experiments.

Contribution

It provides new insights into the effects of rotation and shear on the Tayler instability in MHD flows, including conditions for suppression and destabilization, and explores experimental feasibility.

Findings

Critical Hartmann number independent of Pm for resting cylinders

Rotation suppresses the instability, with maximum quenching at Pm=1

Negative shear destabilizes the toroidal magnetic field

Abstract

The nonaxisymmetric 'kink-type' Tayler instability (TI) of toroidal magnetic fields is studied for conducting incompressible fluids of uniform density between two infinitely long cylinders rotating around the same axis. It is shown that for resting cylinders the critical Hartmann number for the unstable modes does not depend on Pm. By rigid rotation the instability is suppressed where the critical ratio of the rotation velocity and the Alfven velocity of the field (only) slightly depends on the magnetic Prandtl number Pm. For Pm=1 the rotational quenching of TI takes its maximum. Rotation laws with negative shear (i.e. d\Omega/dR<0) strongly destabilize the toroidal field if the rotation is not too fast. For sufficiently high Reynolds numbers of rotation the suppression of the nonaxisymmetric magnetic instability always dominates. The angular momentum transport of the instability is anticorrelated with the shear so that an eddy viscosity can be defined which proves to be positive. For negative shear the Maxwell stress of the perturbations remarkably contributes to the angular momentum transport. We have also shown the possibility of laboratory TI experiments with a wide-gap container filled with fluid metals like sodium or gallium. Even the effect of the rotational stabilization can be reproduced in the laboratory with electric currents of only a few kAmp.